How do you prove #cos ((2pi)/3)#?

Question

Zoe Austin · Accepted Answer

Let's convert to degrees, which are usually easier to work with.

Use the conversion rate #180/pi# to convert to degrees. #=> (2pi)/3 xx 180/pi# #= 120 degrees. We must now determine the reference angle for 120 degrees. Since 120 degrees is in quadrant II, the reference angle is found by using the expression #180 - theta#, where #theta# is the angle in degrees. Calculating we get a reference angle of 60 degrees. We must now apply our knowledge of the special triangles to continue. The special triangle that contains 60 degrees is the 30-60-90 degrees, that has side lengths of #1, sqrt(3) and 2#, respectively. So, the hypotenuse measures 2, the side opposite our angle measures #sqrt(3)# and the side adjacent measures #1#. Applying the definition that cos = adjacent/hypotenuse, we find that our ratio is #1/2#. However, since we're in quadrant II the x axis is negative and therefore our ratio is in fact #-1/2#. Thus, #cos((2pi)/3) = -1/2# You can use the acronym #C-A-S-T (Q. 4-3-2-1)# to remember in which quadrants the ratios are positive. For example, we can say with this acronym that cos is positive in quadrant #IV#. Feel free to ask anything more either on my Socratic dashboard or on the main questions page. I understand that this might at first seem like a long and complicated process. Practice exercises #a) tan300# #b) sin((7pi)/6)# Good luck!